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Physics LibreTexts

8: Sparse Matrices

  • Page ID
    34845
  • A sparse matrix is a matrix in which most of the entries are zero. Such matrices are very commonly encountered in finite-difference equations. For example, when we discretized the 1D Schrödinger wave equation with Dirichlet boundary conditions, we saw that the Hamiltonian matrix had the tridiagonal form

    \[\mathbf{H} = -\frac{1}{2h^2} \begin{bmatrix} -2 & 1 \\ 1 & -2 & \ddots \\ & \ddots & \ddots & 1 \\ & & 1 & -2\end{bmatrix} + \begin{bmatrix}V_0 \\ & V_1 \\& & \ddots \\ & & & V_{N-1}\end{bmatrix}.\]

    Hence, if there are \(N\) diagonalization points, the Hamiltonian matrix has a total of \(N^{2}\) entries, but only \(O(N)\) of these entries are non-zero.

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